## Shortness coefficient of cyclically 4-edge-connected cubic graphs.(English)Zbl 1432.05059

Summary: B. Grünbaum and J. Malkevitch [Aequationes Math. 14, 191–196 (1976; Zbl 0331.05118)] proved that the shortness coefficient of cyclically 4-edge-connected cubic planar graphs is at most $$\frac{76}{77}$$. Recently, this was improved to $$\frac{359}{366}$$ $$(<\frac{52}{53})$$ and the question was raised whether this can be strengthened to $$\frac{41}{42}$$, a natural bound inferred from one of the Faulkner-Younger graphs. We prove that the shortness coefficient of cyclically 4-edge-connected cubic planar graphs is at most $$\frac{37}{38}$$ and that we also get the same value for cyclically 4-edge-connected cubic graphs of genus $$g$$ for any prescribed genus $$g \geqslant 0$$. We also show that $$\frac{45}{46}$$ is an upper bound for the shortness coefficient of cyclically 4-edge-connected cubic graphs of genus $$g$$ with face lengths bounded above by some constant larger than 22 for any prescribed $$g \geqslant 0$$.

### MSC:

 05C40 Connectivity 05C38 Paths and cycles 05C45 Eulerian and Hamiltonian graphs 05C10 Planar graphs; geometric and topological aspects of graph theory

Zbl 0331.05118
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### References:

 [1] R. E. L. Aldred, S. Bau, D. A. Holton, and B. D. McKay.Nonhamiltonian 3connected cubic planar graphs.SIAM J. Discrete Math., 13(1):25-32, 2000. · Zbl 0941.05041 [2] J. A. Bondy and M. Simonovits. Longest cycles in 3-connected 3-regular graphs. Canad. J. Math., 32:987-992, 1980. · Zbl 0454.05043 [3] G. Brinkmann and B. McKay. Fast generation of planar graphs.MATCH Commun. Math. Comput. Chem., 42(4):909-924, 2007.Seehttp://cs.anu.edu.au/  bdm/index.html. [4] G. Brinkmann and C. T. Zamfirescu. Polyhedra with few 3-cuts are hamiltonian. Electr. J. Combin., 26:#P1.39, 2019. · Zbl 1409.05122 [5] G. Chen and X. Yu. Long cycles in 3-connected graphs.J. Combin. Theory, Ser. B, 86(1):80-99, 2002. · Zbl 1025.05036 [6] V. Chv´atal. Flip-flops in hypohamiltonian graphs.Canad. Math. Bull., 16:33-41, 1973. · Zbl 0253.05142 [7] I. Fabrici, J. Harant, T. Madaras, S. Mohr, R. Sot´ak, and C. T. Zamfirescu. Long cycles and spanning subgraphs of locally maximal 1-planar graphs. To appear inJ. Graph Theory. Manuscript available online athttps://arxiv.org/abs/1912.08028, 2018. [8] G. B. Faulkner and D. H. Younger. Non-Hamiltonian cubic planar maps.Discrete Math., 7(1-2):67-74, 1974. · Zbl 0271.05106 [9] H. Fleischner and B. Jackson. A note concerning some conjectures on cyclically 4edge connected 3-regular graphs.Graph Theory in Memory of G. A. Dirac (ed.: L. D. Andersen), Annals Discrete Math., 41:171-177, 1988. [10] E. J. Grinberg. Plane homogeneous graphs of degree three without Hamiltonian circuits (in Russian).Latvian Math. Yearbook, 4:51-58, 1968. · Zbl 0185.27901 [11] B. Gr¨unbaum and J. Malkevitch. Pairs of edge-disjoint Hamiltonian circuits.Aequat. Math., 14(1):191-196, 1976. · Zbl 0331.05118 [12] B. Gr¨unbaum and H. Walther. Shortness exponents of families of graphs.J. Combin. Theory, Ser. A, 14(3):364-385, 1973. · Zbl 0263.05103 [13] J. H¨agglund. On snarks that are far from being 3-edge colorable.Electr. J. Combin., 23:#P2.6, 2016. [14] J. Harant.Uber den Shortness Exponent regul¨¨arer Polyedergraphen mit genau zwei Typen von Elementarfl¨achen (in German). PhD thesis, Technische Hochschule Ilmenau, 1982. [15] J. Harant and H. Walther. Some new results about the shortness exponent in polyhedral graphs.Casopis pro pˇˇestov´an´ı matematiky, 112(2):114-122, 1987. · Zbl 0642.05039 [16] Q. Liu, X. Yu, and Z. Zhang. Circumference of 3-connected cubic graphs.J. Combin. Theory, Ser. B, 128:134-159, 2018. · Zbl 1375.05157 [17] O.-H. S. Lo and J. M. Schmidt. Longest cycles in cyclically 4-edge-connected cubic planar graphs.Australas. J. Combin., 72(1):155-162, 2018. · Zbl 1405.05046 [18] K. Markstr¨om. Improved bounds for the shortness coefficient of cyclically 4-edge connected cubic graphs and snarks. Manuscript available online atarXiv:1309.3870, 2014. [19] E. M´aˇcajov´a and J. Maz´ak. Cubic graphs with large circumference deficit.J. Graph Theory, 82(4):433-440, 2016. [20] J. W. Moon and L. Moser. Simple paths on polyhedra.Pacific J. Math., 13(2):629- 631, 1963. · Zbl 0115.41001 [21] P. J. Owens. Regular planar graphs with faces of only two types and shortness parameters.J. Graph Theory, 8:253-275, 1984. · Zbl 0541.05037 [22] K. Ozeki, N. Van Cleemput, and C. T. Zamfirescu. Hamiltonian properties of polyhedra with few 3-cuts – a survey.Discrete Math., 341(9):2646 - 2660, 2018. · Zbl 1392.05066 [23] E. Steinitz. Polyeder und Raumeinteilungen.Encyklop¨adie der mathematischen Wissenschaften, Band 3 (Geometrie):1-139, 1922. [24] C. Thomassen. A theorem on paths in planar graphs.J. Graph Theory, 7(2):169-176, 1983. · Zbl 0515.05040 [25] C. Thomassen. Reflections on graph theory.J. Graph Theory, 10(3):309-324, 1986. · Zbl 0614.05050 [26] W. T. Tutte. A theorem on planar graphs.Trans. Amer. Math. Soc., 82(1):99-116, 1956. · Zbl 0070.18403 [27] H. Walther. ¨Uber Extremalkreise in regul¨aren Landkarten.Wiss. Z. Techn. Hochsch. Ilmenau, 15:139-142, 1969. · Zbl 0202.23402 [28] H. Walther. Polyhedral graphs without hamiltonian cycles.Discrete Appl. Math., 79(1):257-263, 1997. · Zbl 0883.05092 [29] J. Zaks. Non-hamiltonian simple 3-potytopes having just two types of faces.Discrete Math., 29:87-101, 1980. · Zbl 0445.05065 [30] J. Zaks. Shortness coefficient of cyclically 5-connected cubic planar graphs.Aequat. Math., 25:97-102, 1982. · Zbl 0518.05045 [31] C.
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